Slide rule



Aug. 24, 1965 P. M. PEPPER 3,202,352

SLIDE RULE Filed July 29, 1963 2 Sheets-Sheet 1 HEAR Haw

INVEN TOR.

PAUL M PEPPER z l/ w P. M. PEPPER Aug. 24, 1965 SLIDE RULE 2Sheets-Sheet 2 Filed July 29, 1963 United States Patent i snea'ssz SLIDERULE Paul M. Pepper, 1235 Rustic Place, Columbus 14, Ohio Filed July 29,1963, Ser. No. 298,227

8 Claims. (Cl. 235-83) This invention relates to improvements incalculating devices of the slide rule type and particularly to providingscales for a slide rule that extends its utility, accuracy and greatlysimplifies the computation performed thereon.

There is disclosed in my Patent No. 2,564,227, a slide rule capable ofmaking computations accurate to five significant figures. The rule shownin that patent is of the circular type and has as one of its primaryscales a ten convolution spiral on which is laid out a logarithmicscale. This scale is divided by the secondary logarithmic graduationsinto an equal number of divisions which are ten in number or somemultiple or sub-multiple of ten. A second scale is a ten convolutionspiral logarithmic sine scale. This second scale pertains to sines ofangles in the range from 90 to approximately 5.75 A third scale is a tenconvolution spiral logarithmic tangent scale and pertains to tangents ofangles in the range from 45 to approximately 5.75 For the operationsintended and in these ranges these scales and rule erformed as expected.It can be seen, however, that for computations involving the sine ortangent of an angle below 5.75, approximately ten additionalconvolutions of the appropriate scale are required to get down to 0.575.Beyond this point, the radian measure can be used instead of the sine ortangent without significant loss of accuracy. To put the additional tenturns on the rule shown in that patent or on any other ten turn rulewould require enlarging the size of the rule to unmanageableproportions.

Alternatively, the additional ten turns may be placed on 1 the reverseside of the rule tothe exclusion of other useful scales. Theseadditional ten turns would occupy essentially as much as the entireremaining ranges of angles; i.e., as .much as for angles from 5.75 to 90on the (logarithmic) sine scale and 5.75 to 4-5" on the (logarithmic)tangent scale. Providing a scale with the additional turns would have tobe to the exclusion of such other quite utilitarian scales such as thereversed (logarithmic) number scale (the CI scale), the A scale(half-size cycle logarithmic number scale), the log scales, and the(logarithmic) hyperbolic sine and the cosine scales.

It is the purpose oft-he present invention to extend the logarithmicsine and tangent scales to zero degrees by ;out that range. Forinstance, it is known that for angles between 5.75 and zero degrees, thesine of the angle divided by the angle in degrees is approximately Tidyalso the value of this ratio gets closer to as the angle gets closer tozero. This led to my discovery that settings of the logarithm of one ofthese func- 1 tions .may be obtained from the settings of the otherfunction by an extremely abbreviated scale laid out accordscale.

3,232,352 Patented Aug. 24, 1965 ice ing to the differences of thelogarithms of the functions. In this way, one complete scale of a. firstfunction for the range in quest-ion, together with an abbreviateddifferential scale for a second function in its desired range, aresubstituted for both complete scales of the two functions in theirrespective ranges.

'It is accordingly a principal object of the present invention toprovide a new and improved slide rule that performs computations notpossible with the original slide rule.

It is another object to design an improved slide rule which providescomputations to five significant figures over an extended range.

A further object of the present invention is to increase the utility ofa slide rule to perform computations below 5.75 without appreciablyadding to the physical dimensions or the usable area of the rule.

Another object of the present invention is to compute with values of onefunction from settings of another function on a slide rule without thenecessity of providing a complete scale for the first function.

Other objects and features of the present invention will become apparentfrom the following detailed description when taken in conjunction withthe drawings in which:

FIGURES '1, 1a and 1b are face views of an embodiment of a circularslide rule having a spiral logarithmic scale and a marginal annularscale;

FIGURES 2 and 2a are extensions of the embodiment of FIGURE 1 to includethe features .of the present invention; and,

FIGURE 3 is another embodiment illustrating that the principles of theresent invention are equally applicable to a straight slide rule.

Referring now to FIGURE 1, la and 1b, there is shown a circular sliderule substantially as that disclosed in my prior patent, supra. Withreference thereto, disk 2t) is provided with a marginal annular scale 46divided by equispaced primary numbered graduations 48, the zerograduation so being on the same radius as the radial center of the tab24. A plurality of secondary graduations 52 divide equally andpreferably into ten equal spaces, the portion of the scale betweenadjacent primary graduations 48. A spiral (referred to as a helical inmy prior patent) logarithmic scale 54 having ten convolutions occupiesthe major portion of the physical embodiment of the rule. The outer endof the spiral scale is spaced inwardly from the number scale 46 and theinner end is spaced outwardly from the center 40. The logarithmic scale'54 has its outer end radially aligned with the zero position Stl of thenumber scale 43.

The logarithmic number scale 54 is divided by suitably numbered primaryscale divisions or graduations 56. The spaces between the primarynumbered graduations 56 are divided by the secondary logarithmicgraduations 57 into an equal number of divisions which are ten in numberor some multiple or sub-multiple of ten.

A second spiral scale line 58 is juxtaposed to the scale 54 andpreferably has the same number of convolutions although it may have moreor fewer convolutions than scale 54. One side of this scale may beprovided graduations 60 suitably numbered which constitute a sine-cosinelogarithmic scale. At the opposite side of scale line 58 may be provideda second series of graduations 62 which will preferably comprise atangent-cotangent logarithmic The two sets of logarithmic scales 54, 58are preferably arranged upon the disc, as illustrated in FIGURE 1.

Referring to FIGURE 2, there is illustrated one embodiment of thepresent invention as incorporated in the rule of FIGURE 1. Specifically,there is provided on this rule differential scales 21, 25, 31 and 35 toextend the (logarithmic sine and tangent scales to read with fiveaaoasaa significant figure accuracy of The principle in the design ofthese differential scales is, that for angles close to zero degrees, thesine of the angle is approximately times the number of degrees.approximate ratio gets closer to Also, the value of the as the anglesget closer to zero. Moreover, the difference of the logarithms of thesine and the angle in degrees gets closer to the logarithm of layoutangle 6= log +1.80000 3600 wherein 0 is the layoutangle in degreesclockwise around the'pivotal point measured from the registered hairlineand x is the graduation angle in degrees. Therefore, since a v sinwsinx. IL s1na,x we )where 1snear (andlessthan the differential sin scale21 is designed by finding and laying out the logarithms of these ratiosfor 10 to 1, and the limiting value l for 0 the principle that forangles close to 0 the tangent and the angle in radian measure have aratio of approximately one, or when the number of degrees in angle isused, a 7

ratio of approximately In this particular illustration, the differentialscale 31 forgiving the tangent of angles from 10 to 0 is found by layoutangle 0= log x Therefore, since tan :6

as a setting for a continuation computation; dure of finding or settingthe sine of an angle may be used 41. a differential tan scale can bedesigned by laying out the logarithms of these ratios for 10 to 1 andlimiting value for 0 It will be noted that since the sin x is less thanand the is greater thanv the limiting value p for 0 will yield a commonzero marker for both scales.

in this way the differential sin scale is graduated in a preferredembodiment to the left (counterclockwise) of this common zero markerwhereas the differential tan scale is graduated to the right (clockwise)of this marker.

Utilizing the slide rule of FIGURE 1 having the improvement thereon ofthe differential sin and tan scales 21 and 31 of FIGURE 2 of the presentinvention, the sin of angle may be found to an accuracy of fivesignificant figures by the following procedure: To find the sin of4.37", (a) set the inner arm 24 on register and the outer arm 34 on 437of the (logarithmic) number scale 54, noting the index 0.64 on the outerarm Where its hairline intersects the guideline of the scale 54 at thenumber 437, (b) move the two arms as a unit-without changing the angularrelationship between themuntil the inner arm 24 indicates 4.37 on thedifferential sin scale 21, (c) and, at the sum, 0.88, of the index of0.64 and 0.24, the index of the differential sine scale (at0.88:0.644-024) read sin 4.27=0.076195 from scale 54 by means of thehairline on the outer arm 34. The decimal point was set from theknowledge that sin 4.37 lies between sin 5.75", which is approximately0.1 and sin 0.575, which is approximately 0.01. With this same settingone may also read the log sin 437 taking the last four figures 8193 ofthe mantissa, .88193, from the scale 46, the first figure, 8, of themantissa from thefirst figure of the index 0.88; the characteristic 8..-10 is determined from the decimal point in sin 4.37.. Moreover, sin

may be read from the number scale 54 or may be used This proceeither atthe beginning or internally in a sequence of computations.

The computational procedure may be varied to per-.

form both multiplications and divisions by the angles of the scale.Returning to the previous example, follow step (a) as given above, toread tan 4.37, then shift the two arms 24 and 34 as a unit withoutchanging the angle between them until the lower arm 24 is on 4.37 of thedifferential tan scale and read tan 4.37'=0.076423 or log tan4.37'=8.8832310. It is to be noted that both the sine and tangent can beread for a given angle with a single setting of the angular relation ofthe arms as in (a) above.

To multiply by the sine of an angle, for instance, 8.32 sin 4.37",proceed formally exactly as for an ordinary multiplication of threefactors 8.32 437 S(4.37)=0.63396 where S(4.37) represents 4.37 on thedifferential sine scale. The index 0.24 is used for the third factor andthedecimal point of the (original) product is set from knowing that sin437 is between 0.01 and 0.1.

When the value of sin 2: is given, the formal quotient of this value(set on the number scale 54) by the zero angle on the differential sinscale 20 yields a rough approximation to x to be read on the numberscale 54. A second formal division of sin x on the scale 54 by theroughly determined angle x on the differential sine scale yields areasonably accurate second approximation of the value of x to be readfrom the scale 54. The same procedure would be followed using thedifferential tan scale when the tangent of the angle is given.

It was found more convenient, however, to establish a differential arcsin scale 31 and a differential arc tan scale 35 to perform theconversion in the opposite direction. These differential arc functionscales eliminate the operation of having to determine the approximatevalues of the angle. Since sin x is determinable from the value of sinx, a multiplication of sin x by sin x will yield the value of x, thedesired angle in degrees. To perform this conversion from sin x to x itsuffices in the illustrated embodiment to construct a scale whose layoutangle is given by layout angle 0=360+ (logic 1.80000)3600 sin a:

Graduations on this scale will be preferentially for values of sin xfrom 0.00 to 0.20 as follows:

on the log number scale.

The layout angle for the differential arc tan scale will be given indegrees by layout angle 0=360+ log 1.80000 3600 The scale is graduatedaccording to values of tan x and has the same graduation for tan x=0.00as the differential arc sin scale has for x=0.00. The other graduationswill preferentially be for the following values of tan x:

tan 1 6 0. 01 200. 19 0. 02 20s. 03 0. 03 20s. 71 0. 04 20s. 40

0. 00 207. 37 0. 07 20s. as 0. 08 205. 02 0. 00 205. 03

and labeled according to these values. As in the case of thedifferential sin and tan scales, these also extend in oppositedirections from their common zero graduation.

By observation of the four computed tables of layout angles, it may beseen that the differential sin and tangent scale together occupy on theorder of 24 on a single turn for angles x between 0 and 10 and that thedifferential arc sin and are tangent scales together occupyapproximately 31 on a single turn for tan x and sin x between 0.00 and0.20.

To utilize the (differential) are sin scale 31 or the (differential) arctan scale 35 in converting to an angle, the following procedure would befollowed: To determine the angle x whose tangent is 0.023561, (a) setthe inner arm 24 on register and the outer arm 34 on 23561 of the numberscale 54 at index 0.37, (b) move the two arms 23 and 34- as a unitwithout changing the angle therebetween until the inner arm hairline isat 0.023561 on the arc tan 35 scale, (c) read x=1.3497 at'the inand 0.1.

A similar procedure is used to convert from the sine of the angle to theangle using the arc sine scale.

To find the angle which arises as the sin or tangent of an anglecomputed from a formula, for example,

. rule.

7' the division is performed in the customary way, noting that the valueof the quotient is 0.060971 and then performing the conversion asbefore. Note that it is, not necessary to reset the value of tan x, thequotient, nor to register the inner arm, since the quotient is read orset when this arm is in register. On the rule, (a) shift the .two armsas a unit keeping their angular relationship fixed until the hairline onthe previously registered arm is at 0.061 on the differential arc tanscale, (b) read the angle x=3.04890 from the hairline on the second armat the index 1.54 (or 0.54), the sum of the index 0.78 of the quotient0.06 and the index 0.76 of the arc tan scale.

Although these four diiferential scales are here illustrated as lying oncircular arcs near the periphery of the circular disc, the positions ofthese scales on the rule need not be so limited. The only requirement inpositioning the scales is that the radial lines through the graduationsassume the same angular relations with the register as shown inFIGURE 1. Or alternatively, similar scales may be placed on the armsrather than on the disc andto be used in conjunction with two individualmarks at only r io g and r e 1 180 le T on the disc.

As described herein, the angles .x are measured in degrees; however, asimilar set of differential scales for any other angular measure such asgrads, radians, decimal fractions of a circle, etc., is within the scopeof my invention. Also, some base of logarithms other than may beappropriate for a pair of functions.

The illustrative embodiment shown herein extends the ranges of sines andtangents by means of a set of differential scales applied to a disc-typerule with ten-turn log number and log trigonometric scales, a similarset of scales for a circular slide rule or one for any other number ofturns on either a disc, cylinder, or other surface is also within thescope of my invention. For example,

the layout angles for the differential sine and tangent scales of adisc-type rule with a single cycle logarithmic number scale are given bysin scale: layout angle 0= log x tan scale: layout angle 0= log ZT +2ZGO The layout angles corresponding to the associated difk ferential arcfunction scales are given by are sin scale:

layout angle 0= log 1)360 sin 00 The differential scales are describedabove as having been graduated on a circular slide rule. It is to beexpressly understood that the principles of the present invention may beequally applied to other types of slide For instance, the principles ofthe differential scales have been incorporated in a straight slide rulethroughout the entire range of angles for the sine function extendingfrom 0 to 90.

When the length of. one cycle of the logarithmic number scale is takenas the layout unit, the layout distances 7 D for the graduations of thediiferential sine scales and V differential tangent scales are given bysin :1: 0

sin: D=log tan :2:

g. and those fior the associateddifierential arc sin and are sin x aresin scale: D=log O are tan scale: D=log fi -1 This rule is shown inFIGURE 3. The spread of the graduationswas adequate to give readings andsettings of sine which could be read to three figures throughout .theentire range of angles. The scale is deci-trig, and, although this rulerequires an additional operation for each angle, the differential sinescale, for instance, provides the advantage over the conventional sinescale of being able to read tenths of a degree from 70 to 90 in additionto being extended to read to 0 with accuracy.

Again, it is to be expressly understood that although the abovedifferential scalesare described in conjunction with trigonometricscales, the principles of the invention are not to be limited thereto.For example, the dilference of the function log cosh x and the functionx varies from O to log 2 when x=+ Thus brief differential scales couldbe constructed to read log cosh x from the function x, i.e., the scale46 of FIGURE 1 with equal distances between graduations.

The layout angle for the differential log cosh scale, on the rule ofFIGURE '1, would be given-by 0 (log cosh x-ic) 3600 if five significantfigures are desired, or

0: (log cosh x)360 if four significant figures are sufficient. Thislatter layout angle would apply equally to a circular slide rule of theordinary type having a single turn equal graduation scale.

The layout angle for thecorresponding inverse function scales are givenby (i='(x l0g cosh x) 3600 or (xlog cosh x)360 2: loge cosh x a:

For the ten turn type of rule shown in FIGURE 1, the fiive figures inthe log cosh x -x column are to be multiplied by 3600; and for theconventional circular rule by 360. In a straight slide rule the log coshscale is laid out with the figdires as given in table except 1 is addedto shift the scale a cycle to be used in conjunction with the equaldivision scale.

Finally, and of primary significance, most of the above illustrationsconcerns two functions whose ratio varies slowly, with the consequencethat the difference of their logarithms varies slowly, the same theoryof construction (as shown with the log. cosh x x rule) applies to anytwo functions whose difference varies slowly.

Other modifications and departures from the above preferred embodimentswithin the scope of my invention will be apparent to those skilled inthe art.

What is claimed is:

1. A calculating device of the circular disc type comprising:

(A) a first spiral scale representing a first function extending from apivot point and extending to the outer periphery of said disc;

(B) a second spiral scale juxtaposed to said first spiral scale andrepresenting a second function;

(C) a pair of pivotly mounted arms on said disc and extending from saidpivot point, at least one of said arms extending beyond theouterperiphery of said disc;

(D) a first and a second auxiliary scale positioned on said disc beyondthe outer turn of scale spiral scales,

(1) said first auxiliary scale representing a third function havinggraduations laid out according to the diiferences of said first andthird functions,

(2) said second auxiliary scale representing a fourth function havinggraduations laid out according to the dilferences of said second andfourth functions;

(B) said graduations on said first and second auxiliary scale sopositioned on said disc and angularly related that in determining anunknown when the first of said arms is set at register and the second ofsaid arms is set on one of said spiral scales, and said pair of armsmoved without changing the angle therebtween setting on one of saidauxiliary scales the reading originally set on said spiral scale, saidanswer is read on said spiral scale.

2. A calculating device as set forth in claim 1 wherein said first andsecond spiral scales are logarithmic scales and wherein said graduationsof said first and second auxiliary scale are the differences of thelogarithms of said functions.

3. A calculating device as set forth in claim 1 wherein the ratio of thevalues of said first and third and said second and fourth functionsvaries slowly throughout a given range and wherein the difference of thelogarithms of said two values also varies slowly throughout said range.

10 4. A calculating device as set forth in claim 1 wherein thegraduations on said first auxiliary scale represent the sine of anglesin degrees and are found by layout angle 9=(l0g 1.s0000)3600 5. Acalculating device as set forth in claim 3 wherein the sine of the angledivided by the angle in degrees is approximately and wherein graduationsof said second scale are closer angularly to of said first scale as theangle gets closer to zero.

6. A calculating device as set forth in claim 3 where the graduations onsaid second scale represent the tangent of angles in degrees and arefound by tan 2 layout angle 0 =(1 gl0 1.800%)3600 7. A calculatingdevice as set forth in claim 5 wherein the tangent of the angle dividedby the angle in degrees is approximately and wherein graduations of saidsecond scale are closer angularly to References Cited by the ExaminerUNITED STATES PATENTS 4/ 24 Keulfel 235- 8/51 Pepper 235-67 OTHERREFERENCES Jenkins, Lewis A.: Design of Special Slide Rules I,Industrial Management, vol. 54, November 1917, pages 241-248.

Jenkins, Lewis A.: Design of Special Slide Rules II, IndustrialManagement, December 1917, pages 384-389.

LEO SMILOW, Primary Examiner.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION atent No.3,202,352 August 24, 1965 Paul M, Pepper It is hereby certified thaterror appears in the above numbered patant requiring correction and thatthe said Letters Patent should read as corrected below.

Column 4, line 68, strike out "sin", second occurrence, and insert thesame before "437" in line 69, same column 4; column 6, lines 5 and 6,for

180 read 180 LU TT same column 6, line 53, for "31" read 35 lines 54 and61, for "35", each occurrence, read 25 lines 61 and 63, before "arc",each occurrence, insert differential column 7, line 50, for "260" read360 column 8, line 31, for "6=(log cosh X-)360" read E':)=(log coshx-X)360 line 45, after "to" insert 7.0 column 10, lines 5 and 6 andlines 24 and 25, before "1 ,80000", each occurrence,

insert Signed and sealed this 14th day of June 1966.

(SEAL) Attest:

ERNEST W. SWIDER EDWARD J. BRENNER Attesting Officer Commissioner ofPatents

1. A CALCULATING DEVICE OF THE CIRCULAR DISC TYPE COMPRISING: (A) AFIRST SPIRAL SCALE REPRESENTING A FIRST FUNCTION EXTENDING FROM A PIVOTPOINT AND EXTENDING TO THE OUTER PERIPHERY OF SAID DISC; (B) A SECONDSPIRAL SCALE JUXTAPOSED TO SAID FIRST SPIRAL SCALE AND REPRESENTING ASECOND FUNCTION; (C) A PAIR OF PIVOTLY MOUNTED ARMS ON SAID DISC ANDEXTENDING FROM SAID PIVOT POINT, AT LEAST ONE OF SAID ARMS EXTENDINGBEYOND THE OUTER PERIPHERY OF SAID DISC; (D) A FIRST AND A SECONDAUXILIARY SCALE POSITIONED ON SAID DISC BEYOND THE OUTER TURN OF SCALESPIRAL SCALES, (1) SAID FIRST AUXILIARY SCALE REPRESENTING A THIRDFUNCTION HAVING GRADUATIONS LAID OUT ACCORDING TO THE DIFFERENCES OFSAID FIRST AND THIRD FUNCTIONS, (2) SAID SECOND AUXILIARY SCALEREPRESENTING A FOURTH FUNCTION HAVING GRADUATIONS LAID OUT ACCORDING TOTHE DIFFERENCES OF SAID SECOND AND FOURTH FUNCTIONS;